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<p>In the article a class of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation>-valued monogenic fractional power functions defined in the reduced quaternions and depending on the parameters <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(p\in \mathbb {N}_{0}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and real <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\lambda &gt; -1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>&gt;</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> is constructed. These functions are an extension of the well-known class of orthogonal Appell polynomials, which is included as a special case. For the monogenic fractional powers essential properties, i.e. monogenicity, a generalized Appell property and a two-step recurrence formula, are proved and their corresponding Kelvin transforms in terms of a corresponding anti-monogenic fractional power function are given.</p>

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Monogenic Fractional Powers in \(\mathbb {H}\)

  • 期刊论文

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  • 发表日期：

    2025年10月28日

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  • Sebastian Bock

摘要

In the article a class of \(\mathbb {H}\) H -valued monogenic fractional power functions defined in the reduced quaternions and depending on the parameters \(p\in \mathbb {N}_{0}\) p ∈ N 0 and real \(\lambda > -1\) λ > - 1 is constructed. These functions are an extension of the well-known class of orthogonal Appell polynomials, which is included as a special case. For the monogenic fractional powers essential properties, i.e. monogenicity, a generalized Appell property and a two-step recurrence formula, are proved and their corresponding Kelvin transforms in terms of a corresponding anti-monogenic fractional power function are given.